1 Answer
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Causative has two meanings: forcing and allowing. Let me first illustrate the semantic distinction considering only "forcing":

Negation of causative of P = "not force doing P"

Causative of negation of P = "force to not do P"

In general, there is a mismatch between the hierarchy in the syntactic structure (at the surface) and the hierarchy in the logical structure. For example, the English expression I cannot do P does not mean "It is possible ('can') for me to not do P" (potential of negation of P) but means "It is not the case that I can do P" (negation of potential of P). What the article is probably saying is that tabe-sase-nai syntactically looks like the hierarchy is "negation of causative of eat", but its meaning is "causative of negation of eat", or "forcing not to eat".

Now, there is the same semantic distinction for "allowing":

Negation of causative of P = "not (particularly) allow doing P"

Causative of negation of P = "allow not doing P"

Note that "forcing" and "allowing" are logically connected under the following
formulae from modal logic (~ = negation, ◇ = possibility operator (i.e., "allowing" in this case), □ = necessity operator (i.e., "forcing" in this case)):

~◇P = □~P
~□P = ◇~P

From this, the expression tabe-sase-nai which means "forcing not to eat" can also be paraphrased as "not allowing to eat".

Can you clarify the itemized list at the beginning? To me, “not allow doing P” exactly means “force not to do P,” and “not force doing P” exactly means “allow not doing P,” so the correspondence among the four in the list is not what I expect.
– Tsuyoshi ItoJun 21 '12 at 13:09

@TsuyoshiIto Thanks for pointing that out. I somehow had it the other way around. I hope it makes sense now.
– user458Jun 21 '12 at 13:47

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Thanks. Actually my confusion had another layer, but now I think I understand the correspondence. You are referring to two meanings of causative (“allow” and “force”), and you construct negation of causative and causative of negation in both meanings. The difficult point for me was that “not force doing P” exactly looks like the same thing as “allow not doing P.” (If you do not force me to swim, I am allowed not to swim.) It might become easier to understand if you focus on one of the two meanings of causative first, and then add an explanation that causative can sometimes mean the other.
– Tsuyoshi ItoJun 21 '12 at 13:58

@TsuyoshiIto You are right. I packed two meanings into one paradigm. "Force" is related to the necessity operator □, and "allow" is related to the possibility operator ◇. I actually had a bit of concern that it might be misleading. And that let me end up writing the formulae from modal logic. Let me try to edit again.
– user458Jun 21 '12 at 14:05

@sawa Your last paragraph states that the expression "tabe-sase-nai" means "forcing you not to eat". But doesn't "tabe-sase-nai" mean "not forcing you to eat" ?
– PacerierJun 21 '12 at 19:41